cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A168072 Expansion of (1-27x^2-108x^3)/((1-3x)^2*(1+3x+9x^2)).

Original entry on oeis.org

1, 3, -18, -135, -405, -1944, -8019, -24057, -91854, -334611, -1003833, -3542940, -12223143, -36669429, -124357194, -416118303, -1248354909, -4132485216, -13559717115, -40679151345, -132497807238, -428874481323, -1286623443969, -4142299868388
Offset: 0

Views

Author

Paul Barry, Nov 18 2009

Keywords

Comments

Hankel transform of A168073.

Links

Crossrefs

Cf. A168071, A168073.

Programs

  • Magma
    I:=[1,3,-18,-135]; [n le 4 select I[n] else 3*Self(n-1)+27*Self(n-3)-81*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jul 10 2016
  • Mathematica
    LinearRecurrence[{3, 0, 27, -81}, {1, 3, -18, -135}, 50] (* G. C. Greubel, Jul 09 2016 *)
CoefficientList[Series[(1 - 27 x^2 - 108 x^3) / ((1 - 3 x)^2 (1 + 3 x + 9 x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, Jul 10 2016 *)
  • PARI
    Vec((1-27*x^2-108*x^3)/((1-3*x)^2*(1+3*x+9*x^2)) + O(x^30)) \\ Michel Marcus, Dec 03 2014

Formula

a(n) = 3^n*A168071(n).
a(n) = 3^n*(sin(2*Pi*n/3)/sqrt(3) - cos(2*Pi*n/3) - 2*n + 2). - Ilya Gutkovskiy, Jul 10 2016

Extensions

Corrected by R. J. Mathar, Dec 03 2014

A168076 Expansion of 1 - 3*(1-x-sqrt(1-2*x-3*x^2))/2.

Original entry on oeis.org

1, 0, -3, -3, -6, -12, -27, -63, -153, -381, -969, -2505, -6564, -17394, -46533, -125505, -340902, -931716, -2560401, -7070337, -19609146, -54597852, -152556057, -427642677, -1202289669, -3389281245, -9578183391, -27130207503, -77009455428, -219023318406
Offset: 0

Views

Author

Paul Barry, Nov 18 2009

Keywords

Comments

For n>0, a(n) = -3*A168049(n). Hankel transform is A168075. Another variant is A168073.

Links

Crossrefs

Cf. A168049, A168073, A168075.

Programs

  • Maple
    f:= gfun:-rectoproc({(3*n-3)*a(n)+(1+2*n)*a(n+1)+(-n-2)*a(n+2), a(0) = 1, a(1) = 0, a(2) = -3},a(n),remember):
map(f, [$0..60]); # Robert Israel, May 13 2018
  • Mathematica
    CoefficientList[Series[1 - 3*(1 - x - Sqrt[1 - 2*x - 3*x^2])/2, {x,0,50}] , x] (* G. C. Greubel, Jul 09 2016 *)
  • PARI
    x='x+O('x^99); Vec(1-3*(1-x-(1-2*x-3*x^2)^(1/2))/2) \\ Altug Alkan, May 13 2018
  • PARI
    A168076(n)=!n-3*sum(k=0,n\2-1, binomial(n-2,2*k)*binomial(2*k,k)/(k+1)) \\ M. F. Hasler, May 13 2018

Formula

a(n) = 0^n - 3*Sum_{k=0..floor((n-2)/2), C(n-2,2k)*A000108(k)}.
D-finite with recurrence: n*a(n) + (-2*n+3)*a(n-1) + 3*(-n+3)*a(n-2) = 0. - R. J. Mathar, Dec 03 2014
Recurrence (for n >= 3) follows from the differential equation (3*x^2+2*x-1)*y' - (3*x+1)*y = 3*x-1 satisfied by the g.f. - Robert Israel, May 13 2018
a(n) ~ -3^(n+1/2) / (2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Dec 03 2014
a(n) = -A168073(n) <= 0 for n >= 1. - M. F. Hasler, May 13 2018

Extensions

Comment corrected by Vaclav Kotesovec, Dec 03 2014
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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